View source for Volume

{{Short description|Quantity of three-dimensional space}}
{{Other uses}}
{{CS1 config|mode=cs1}}
{{Infobox physical quantity
| name = Volume
| image = Simple Measuring Cup.jpg
| caption = A [[measuring cup]] can be used to measure volumes of [[liquid]]s. This cup measures volume in units of [[Cup (unit)|cups]], [[fluid ounce]]s, and [[millilitre]]s.
| unit = [[cubic metre]]
| otherunits = [[Litre]], [[fluid ounce]], [[gallon]], [[quart]], [[pint]], [[teaspoon|tsp]], [[dram (unit)|fluid dram]], [[cubic inch|in<sup>3</sup>]], [[cubic yard|yd<sup>3</sup>]], [[Barrel (unit)|barrel]]
| symbols = ''V''
| baseunits = [[metre|m]]<sup>3</sup>
| dimension = '''L'''<sup>3</sup>
|extensive=yes
|intensive=no
|conserved=yes for [[solid]]s and [[liquid]]s, no for [[gas]]es, and [[Plasma (physics)|plasma]]{{efn|At constant temperature and pressure, ignoring other states of matter for brevity}}
|transformsas=conserved}}

'''Volume''' is a [[Measure (mathematics)|measure]] of [[Region (mathematics)|regions]] in [[three-dimensional space]].<ref name="NIST-2022">{{Cite journal |date=April 13, 2022 |title=SI Units - Volume |url=https://www.nist.gov/pml/owm/si-units-volume |journal=[[National Institute of Standards and Technology]] |access-date=August 7, 2022 |archive-date=August 7, 2022 |archive-url=https://web.archive.org/web/20220807105244/https://www.nist.gov/pml/owm/si-units-volume |url-status=live }}</ref> It is often quantified numerically using [[SI derived unit]]s (such as the [[cubic metre]] and [[litre]]) or by various [[imperial units|imperial]] or [[United States customary units|US customary units]] (such as the [[gallon]], [[quart]], [[cubic inch]]). The definition of [[length]] and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of [[fluid]] (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. 
By [[metonymy]], the term "volume" sometimes is used to refer to the corresponding region (e.g., [[bounding volume]]).<ref>{{cite web | title=IEC 60050 — Details for IEV number 102-04-40: "volume" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-04-40 | language=ja | access-date=2023-09-19}}</ref><ref>{{cite web | title=IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-04-39 | language=ja | access-date=2023-09-19}}</ref>

In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple [[Three dimensional|three-dimensional]] shapes can have their volume easily calculated using [[arithmetic]] [[formula]]s. Volumes of more complicated shapes can be calculated with [[integral calculus]] if a formula exists for the shape's boundary. [[Zero-dimensional space|Zero-]], [[One-dimensional space|one-]] and [[two-dimensional]] objects have no volume; in [[Four-dimensional space|four]] and higher dimensions, an analogous concept to the normal volume is the hypervolume.

== History ==
=== Ancient history ===
[[File:Pompeji_6_Hohlmaße_aus_Glas.jpg|thumb|6&nbsp;volumetric measures from the ''mens ponderia'' in [[Pompeii]], an ancient municipal institution for the control of weights and measures]]

The precision of volume measurements in the ancient period usually ranges between {{Cvt|10–50|mL|USoz impoz|sigfig=1}}.<ref name="Imhausen-2016" />{{Rp|page=8}} The earliest evidence of volume calculation came from [[ancient Egypt]] and [[Mesopotamia]] as mathematical problems, approximating volume of simple shapes such as [[cuboid]]s, [[cylinder]]s, [[frustum]] and [[cone]]s. These math problems have been written in the [[Moscow Mathematical Papyrus]] (c. 1820 BCE).<ref name="Treese-2018" />{{Rp|page=403}} In the [[Reisner Papyrus]], ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material.<ref name="Imhausen-2016">{{Cite book |last=Imhausen |first=Annette |url= |title=Mathematics in Ancient Egypt: A Contextual History |date=2016 |publisher=[[Princeton University Press]] |isbn=978-1-4008-7430-9 |location= |oclc=934433864}}</ref>{{Rp|page=116}} The Egyptians use their units of length (the [[cubit]], [[Palm (unit)|palm]], [[Digit (unit)|digit]]) to devise their units of volume, such as the volume cubit<ref name="Imhausen-2016" />{{Rp|page=117}} or deny<ref name="Treese-2018" />{{Rp|page=396}} (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit).<ref name="Imhausen-2016" />{{Rp|page=117}}

The last three books of [[Euclid's Elements|Euclid's ''Elements'']], written in around 300 BCE, detailed the exact formulas for calculating the volume of [[parallelepiped]]s, cones, [[pyramid]]s, cylinders, and [[sphere]]s. The formula were determined by prior mathematicians by using a primitive form of [[Integral|integration]], by breaking the shapes into smaller and simpler pieces.<ref name="Treese-2018">{{Cite book |last=Treese |first=Steven A. |title=History and Measurement of the Base and Derived Units |date=2018 |publisher=[[Springer Science+Business Media]] |isbn=978-3-319-77577-7 |location=Cham, Switzerland |lccn=2018940415 |oclc=1036766223}}</ref>{{Rp|page=403}} A century later, [[Archimedes]] ({{Circa|287 – 212 BCE}}) devised approximate volume formula of several shapes using the [[method of exhaustion]] approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by [[Liu Hui]] in the 3rd century CE, [[Zu Chongzhi]] in the 5th century CE, the [[Middle East]] and [[India]].<ref name="Treese-2018" />{{Rp|page=404}}

Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object.<ref name="Treese-2018" />{{Rp|page=404}} Though highly popularized, Archimedes probably does not submerge the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved.<ref>{{cite web |last=Rorres |first=Chris |title=The Golden Crown |url=http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html |url-status=live |archive-url=https://web.archive.org/web/20090311051318/http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html |archive-date=11 March 2009 |access-date=24 March 2009 |publisher=[[Drexel University]]}}</ref> Instead, he likely have devised a primitive form of a [[Hydrostatic Balance|hydrostatic balance]]. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a [[weighing scale]] submerged underwater, which will tip accordingly due to the [[Archimedes' principle]].<ref>{{Cite journal |last=Graf |first=E. H. |date=2004 |title=Just what did Archimedes say about buoyancy? |url=https://aapt.scitation.org/doi/10.1119/1.1737965 |journal=The Physics Teacher |volume=42 |issue=5 |pages=296–299 |bibcode=2004PhTea..42..296G |doi=10.1119/1.1737965 |access-date=2022-08-07 |archive-date=2021-04-14 |archive-url=https://web.archive.org/web/20210414102422/https://aapt.scitation.org/doi/10.1119/1.1737965 |url-status=live }}</ref>

=== Calculus and standardization of units ===
{{Further|History of calculus|Apothecaries' system}}
[[File:"How to Measure" diagram, with graduated cylinder measuring fluid drams, 1926.jpg|alt=Pouring liquid to a marked flask|left|thumb|Diagram showing how to measure volume using a graduated cylinder with [[fluid dram]] markings, 1926]]
In the [[Middle Ages]], many units for measuring volume were made, such as the [[sester]], [[Amber (unit)|amber]], [[Coomb (unit)|coomb]], and [[Seam (unit)|seam]]. The sheer quantity of such units motivated British kings to standardize them, culminated in the [[Assize of Bread and Ale]] statute in 1258 by [[Henry III of England]]. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel.<ref name="Imhausen-2016" />{{Rp|page=|pages=73–74}} In 1618, the ''[[London Pharmacopoeia]]'' (medicine compound catalog) adopted the Roman gallon<ref name="RPS-2020">{{Cite web |date=4 Feb 2020 |title=Balances, Weights and Measures |url=https://www.rpharms.com/Portals/0/MuseumLearningResources/11%20Balances%20Weights%20and%20Measures.pdf |access-date=13 August 2022 |website=[[Royal Pharmaceutical Society]] |page=1 |archive-date=20 May 2022 |archive-url=https://web.archive.org/web/20220520094140/https://www.rpharms.com/Portals/0/MuseumLearningResources/11%20Balances%20Weights%20and%20Measures.pdf |url-status=live }}</ref> or ''[[congius]]''<ref>{{Cite book |last=Cardarelli |first=François |title=Scientific Unit Conversion: A Practical Guide to Metrication |date=6 Dec 2012 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4471-0805-4 |edition=2nd |location=London |pages=151 |oclc=828776235}}</ref> as a basic unit of volume and gave a conversion table to the apothecaries' units of weight.<ref name="RPS-2020" /> Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between {{Cvt|1–5|mL|USoz impoz|sigfig=1}}.<ref name="Imhausen-2016" />{{Rp|page=8}}

Around the early 17th century, [[Bonaventura Cavalieri]] applied the philosophy of modern integral calculus to calculate the volume of any object. He devised [[Cavalieri's principle]], which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by [[Pierre de Fermat]], [[John Wallis]], [[Isaac Barrow]], [[James Gregory (mathematician)|James Gregory]], [[Isaac Newton]], [[Gottfried Wilhelm Leibniz]] and [[Maria Gaetana Agnesi]] in the 17th and 18th centuries to form the modern integral calculus, which remains in use in the 21st century.<ref name="Treese-2018" />{{Rp|page=404}}

=== Metrication and redefinitions ===
{{Further|History of the metric system}}

On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the ''[[stère]]''&nbsp;(1&nbsp;m<sup>3</sup>) for volume of firewood; the ''[[litre]]''&nbsp;(1&nbsp;dm<sup>3</sup>) for volumes of liquid; and the ''[[gram]]me'', for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice.<ref>
{{cite thesis
 | last = Cox | first = Edward Franklin
 | id = {{ProQuest|301905667}}
 | pages = 99–100
 | publisher = Indiana University
 | title = A History of the Metric System of Weights and Measures, with Emphasis on Campaigns for its Adoption in Great Britain, and in The United States Prior to 1914
 | type = PhD thesis
 | year = 1958
}}</ref> Thirty years later in 1824, the [[imperial gallon]] was defined to be the volume occupied by ten [[Pound (mass)|pounds]] of water at {{Cvt|62|F|C|order=flip}}.<ref name="Treese-2018" />{{Rp|page=394}} This definition was further refined until the United Kingdom's [[Weights and Measures Act 1985]], which makes 1 imperial gallon precisely equal to 4.54609&nbsp;litres with no use of water.<ref>{{Cite book |last=Cook |first=James L. |url= |title=Conversion Factors |date=1991 |publisher=[[Oxford University Press]] |isbn=0-19-856349-3 |location=Oxford [England] |pages=xvi |oclc=22861139}}</ref>

The 1960 redefinition of the metre from the [[International Prototype Metre]] to the orange-red [[Spectral line|emission line]] of [[krypton-86]] atoms unbounded the metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre.<ref>{{cite book |last=Marion |first=Jerry B. |title=Physics For Science and Engineering |publisher=CBS College Publishing |year=1982 |isbn=978-4-8337-0098-6 |page=3}}</ref> The definition of the metre was redefined again in 1983 to use the [[speed of light]] and [[second]] (which is derived from the [[caesium standard]]) and [[2019 revision of the SI|reworded for clarity in 2019]].<ref>{{Cite web |date=20 May 2019 |title=''Mise en pratique'' for the definition of the metre in the SI |url=https://www.bipm.org/documents/20126/41489670/SI-App2-metre.pdf |website=[[International Bureau of Weights and Measures]] |publisher=Consultative Committee for Length |pages=1 |access-date=13 August 2022 |archive-date=13 August 2022 |archive-url=https://web.archive.org/web/20220813164032/https://www.bipm.org/documents/20126/41489670/SI-App2-metre.pdf |url-status=live }}</ref>

== Properties ==
{{Further|Volume element|Volume form}}
As a [[Measure (mathematics)|measure]] of the [[three-dimensional space|Euclidean three-dimensional space]], volume cannot be physically measured as a negative value, similar to [[length]] and [[area]]. Like all continuous [[Monotonic function|monotonic]] (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to [[Cavalieri's principle]] and to the [[infinitesimal calculus]] of three-dimensional bodies.<ref>{{Cite web |title=Volume - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Volume |access-date=2023-05-27 |website=encyclopediaofmath.org}}</ref> A 'unit' of infinitesimally small volume in integral calculus is the [[volume element]]; this formulation is useful when working with different [[Coordinate system|coordinate systems]], spaces and [[Manifold|manifolds]].

== Measurement ==
The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and [[Pinch (action)|pinches]]. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable [[container]]s found in nature, such as [[gourd]]s, sheep or pig [[stomach]]s, and [[Urinary bladder|bladders]]. Later on, as [[metallurgy]] and [[glass production]] improved, small volumes nowadays are usually measured using standardized human-made containers.<ref name="Treese-2018" />{{Rp|page=393}} This method is common for measuring small volume of fluids or [[granular material]]s, by using a [[Multiple (mathematics)|multiple]] or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure [[cooking ingredient]]s.<ref name="Treese-2018" />{{Rp|page=399}}

[[Air displacement pipette]] is used in [[biology]] and [[biochemistry]] to measure volume of fluids at the microscopic scale.<ref>{{cite web |title=Use of Micropipettes |url=http://faculty.buffalostate.edu/wadswogj/courses/bio211%20page/resources/micropipetting%20lab.pdf |url-status=dead |archive-url=https://web.archive.org/web/20160804033455/http://faculty.buffalostate.edu/wadswogj/courses/bio211%20page/resources/micropipetting%20lab.pdf |archive-date=4 August 2016 |accessdate=19 June 2016 |website=[[Buffalo State College]]}}</ref> Calibrated [[measuring cup]]s and [[Measuring spoon|spoons]] are adequate for cooking and daily life applications, however, they are not precise enough for [[laboratory|laboratories]]. There, volume of liquids is measured using [[graduated cylinder]]s, [[pipette]]s and [[volumetric flask]]s. The largest of such calibrated containers are petroleum [[storage tank]]s, some can hold up to {{Cvt|1000000|oilbbl|L|lk=in|abbr=off}} of fluids.<ref name="Treese-2018" />{{Rp|page=399}} Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made.<ref name="Treese-2018" />{{Rp|page=403}}

For even larger volumes such as in a [[reservoir]], the container's volume is modeled by shapes and calculated using mathematics.<ref name="Treese-2018" />{{Rp|page=403}}

=== Units ===
{{main|Unit of volume|Orders of magnitude (volume)}}
[[File:Visualisation litre gram.svg|thumb|Some SI units of volume to scale and approximate corresponding mass of water]]

To ease calculations, a unit of volume is equal to the volume occupied by a [[unit cube]] (with a side length of one). Because the volume occupies three dimensions, if the [[metre]] (m) is chosen as a unit of length, the corresponding unit of volume is the [[cubic metre]] (m<sup>3</sup>). The cubic metre is also a [[SI derived unit]].<ref>{{Cite journal |date=February 25, 2022 |title=Area and Volume |url=https://www.nist.gov/pml/owm/area-and-volume |journal=[[National Institute of Standards and Technology]] |access-date=August 7, 2022 |archive-date=August 7, 2022 |archive-url=https://web.archive.org/web/20220807105300/https://www.nist.gov/pml/owm/area-and-volume |url-status=live }}</ref> Therefore, volume has a [[Dimensional analysis|unit dimension]] of L<sup>3</sup>.<ref>{{Cite book |last=Lemons |first=Don S. |title=A Student's Guide to Dimensional Analysis |date=16 March 2017 |publisher=[[Cambridge University Press]] |isbn=978-1-107-16115-3 |location=New York |page=38 |oclc=959922612}}</ref>

The metric units of volume uses [[metric prefix]]es, strictly in [[Power of 10|powers of ten]]. When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3&nbsp;cm<sup>3</sup> = 2.3 (cm)<sup>3</sup> = 2.3 (0.01 m)<sup>3</sup> = 0.0000023 m<sup>3</sup> (five zeros).<ref name="IBWM-2019">{{SIbrochure9th}}</ref>{{Rp|page=143}}

Commonly used prefixes for cubed length units are the cubic millimetre (mm<sup>3</sup>), cubic centimetre (cm<sup>3</sup>), cubic decimetre (dm<sup>3</sup>), cubic metre (m<sup>3</sup>) and the cubic kilometre (km<sup>3</sup>). The conversion between the prefix units are as follows: 1000&nbsp;mm<sup>3</sup> = 1&nbsp;cm<sup>3</sup>, 1000&nbsp;cm<sup>3</sup> = 1&nbsp;dm<sup>3</sup>, and 1000&nbsp;dm<sup>3</sup> = 1&nbsp;m<sup>3</sup>.<ref name="NIST-2022" /> The [[metric system]] also includes the [[litre]] (L) as a unit of volume, where 1 L = 1&nbsp;dm<sup>3</sup> = 1000&nbsp;cm<sup>3</sup> = 0.001&nbsp;m<sup>3</sup>.<ref name="IBWM-2019" />{{Rp|page=145}} For the litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000&nbsp;mL = 1&nbsp;L, 10&nbsp;mL = 1&nbsp;cL, 10&nbsp;cL = 1&nbsp;dL, and 10&nbsp;dL = 1&nbsp;L.<ref name="NIST-2022" />

Various other [[Imperial units|imperial]] or [[United States customary units|U.S. customary]] units of volume are also in use, including:<ref name="Treese-2018" />{{Rp|page=|pages=396–398}}
* [[cubic inch]], [[cubic foot]], [[cubic yard]], [[acre-foot]], [[cubic mile]];
* [[minim (unit)|minim]], [[Dram (unit)|drachm]], [[fluid ounce]], [[pint]];
* [[teaspoon]], [[tablespoon]];
* [[gill (volume)|gill]], [[quart]], [[gallon]], [[barrel (unit)|barrel]];
* [[cord (unit)|cord]], [[peck]], [[bushel]], [[hogshead]].

=== Capacity and volume ===
Capacity is the maximum amount of material that a container can hold, measured in volume or [[weight]]. However, the contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a {{Cvt|50000|oilbbl|L}} tank that can just hold {{Cvt|7200|MT|lb}} of [[fuel oil]] will not be able to contain the same {{Cvt|7200|MT|lb}} of [[naphtha]], due to naphtha's lower density and thus larger volume.<ref name="Treese-2018" />{{Rp|page=|pages=390–391}}

== Computation ==

=== Basic shapes ===
[[File:visual_proof_cone_volume.svg|thumb|[[Proof without words]] that the volume of a cone is a third of a cylinder of equal diameter and height
{|
|valign="top"|{{nowrap|1.}}||A cone and a cylinder have {{nowrap|radius ''r''}} and {{nowrap|height ''h''.}}
|-
|valign="top"|2.||The volume ratio is maintained when the height is scaled to {{nowrap|1=''h' ''= ''r'' &#8730;{{pi}}.}}
|-
|valign="top"|3.||Decompose it into thin slices.
|-
|valign="top"|4.||Using Cavalieri's principle, reshape each slice into a square of the same area.
|-
|valign="top"|5.||The pyramid is replicated twice.
|-
|valign="top"|6.||Combining them into a cube shows that the volume ratio is 1:3.
|}]]
{{See also|List of formulas in elementary geometry}}For many shapes such as the [[cube]], [[cuboid]] and [[cylinder]], they have an essentially the same volume calculation formula as one for the [[Prism (geometry)|prism]]: the [[Base (geometry)|base]] of the shape multiplied by its [[height]].

=== Integral calculus ===
{{Further|Volume integral}}[[File:Integral_apl_rot_objem3.svg|alt=f(x) and g(x) rotated in the x-axis|thumb|Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x).]]
The calculation of volume is a vital part of [[integral]] calculus. One of which is calculating the volume of [[Solid of revolution|solids of revolution]], by rotating a [[plane curve]] around a [[Line (geometry)|line]] on the same plane. The washer or [[disc integration]] method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:<math display="block">V = \pi \int_a^b \left| f(x)^2 - g(x)^2\right|\,dx</math>where <math display="inline">f(x)</math> and <math display="inline">g(x)</math> are the plane curve boundaries.<ref name="RIT-2014">{{Cite web |date=22 September 2014 |title=Volumes by Integration |url=https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/C8_VolumesbyIntegration_BP_9_22_14.pdf |url-status=live |archive-url=https://web.archive.org/web/20220202194113/https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/C8_VolumesbyIntegration_BP_9_22_14.pdf |archive-date=2 February 2022 |access-date=12 August 2022 |website=[[Rochester Institute of Technology]] }}</ref>{{Rp|pages=1,3}} The [[shell integration]] method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:<ref name="RIT-2014"/>{{Rp|pages=6}}<math display="block">V = 2\pi \int_a^b x |f(x) - g(x)|\, dx</math> The volume of a [[region (mathematics)|region]] ''D'' in [[three-dimensional space]] is given by the triple or [[volume integral]] of the constant [[function (mathematics)|function]] <math>f(x,y,z) = 1</math> over the region. It is usually written as:<ref>{{cite book |last=Stewart |first=James |url=https://archive.org/details/calculusearlytra00stew_1 |title=Calculus: Early Transcendentals |date=2008 |publisher=Brooks Cole Cengage Learning |isbn=978-0-495-01166-8 |edition=6th |author-link=James Stewart (mathematician) |url-access=registration}}</ref>{{rp|at=Section 14.4}}
<math display="block">\iiint_D 1 \,dx\,dy\,dz.</math>

In [[cylindrical coordinate system|cylindrical coordinates]], the [[volume integral]] is
<math display="block">\iiint_D r\,dr\,d\theta\,dz, </math>

In [[spherical coordinate system|spherical coordinates]] (using the convention for angles with <math>\theta</math> as the azimuth and <math>\varphi</math> measured from the polar axis; see more on [[Spherical coordinate system#Conventions|conventions]]), the volume integral is
<math display="block">\iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .</math>

=== Geometric modeling ===
[[File:Dolphin_triangle_mesh.png|alt=Tiled triangles to form a dolphin shape|thumb|250x250px|[[Low poly]] triangle mesh of a dolphin]]
A [[polygon mesh]] is a representation of the object's surface, using [[polygon]]s. The [[volume mesh]] explicitly define its volume and surface properties.

== Derived quantities ==
{{See also|List of physical quantities}}

* [[Density]] is the substance's [[mass]] per unit volume, or total mass divided by total volume.<ref>{{Cite web |last=Benson |first=Tom |date=7 May 2021 |title=Gas Density |url=https://www.grc.nasa.gov/WWW/BGH/fluden.html |access-date=2022-08-13 |website=[[Glenn Research Center]] |archive-date=2022-08-09 |archive-url=https://web.archive.org/web/20220809085244/https://www.grc.nasa.gov/WWW/BGH/fluden.html |url-status=live }}</ref>
* [[Specific volume]] is total volume divided by mass, or the inverse of density.<ref>{{Cite book |last1=Cengel |first1=Yunus A. |url=https://archive.org/details/thermodynamicsen00ceng_0/page/11 |title=Thermodynamics: an engineering approach |last2=Boles |first2=Michael A. |publisher=[[McGraw-Hill]] |year=2002 |isbn=0-07-238332-1 |location=Boston |pages=11 |url-access=registration}}</ref>
* The [[volumetric flow rate]] or [[Discharge (hydrology)|discharge]] is the volume of fluid which passes through a given surface per unit time.
* The [[volumetric heat capacity]] is the [[heat capacity]] of the substance divided by its volume.

== See also ==

* [[Banach–Tarski paradox]]
* [[Dimensional weight]]
* [[Dimensioning]]

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== External links ==
{{Commons category}}
* {{wikibooks-inline|Geometry|Chapter 8|Perimeters, Areas, Volumes}}
* {{wikibooks-inline|Calculus|Volume}}

{{Authority control}}

[[Category:Volume| ]]